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The precise determination of the forming limit strains of a sheet metal is necessary in order to accurately predict the onset of failure in sheet metal forming processes. The method used to detect the onset of necking (i.e. the necking criterion) is a most important factor in formability analysis as it significantly affects the forming limit strains. Many different necking criteria have been developed, which take advantage of the flexibility and high resolution of digital image correlation (DIC) strain measurements. Several time-dependent and time/geometry-dependent necking criteria were carefully investigated in order to evaluate their ability to reliably and consistently detect the onset of necking of TRIP780 sheet specimens that were stretch-formed over a hemispherical punch. It was found that the geometry-dependent, surface slope criterion was the most robust and consistent criterion of those evaluated.

In a number of sheet metal stamping and joining processes, the material undergoes large plastic deformation exceeding the range of plastic strain achievable in a standard tensile test prior to material plastic instability. In order to extend the range of effective strains, the multistep rolling process was employed which enabled prestraining of aluminum sheet above 2.0 of true strain. Tensile testing of rolled samples was used to identify the flow stress corresponding to the level of prestrain: By varying the prestrain level, several data points were obtained for the studied flow curve. The numerical simulation using Abaqus software for the cold rolling process of aluminum strips confirmed that majority of the strip is deformed in plane strain compression condition. Performed simulation of the LDH test determined that earlier fracture might occur if the curve obtained via rolling–tensile testing approach is used versus traditional power law approximation and Voce law approximation. The results of simulation for the multistep drawing of a cylindrical cup revealed possible wrinkling in the die entry area during redrawing stage of the process if the rolling–tensile testing flow curve is employed.

Numerical simulations of uniaxial tensile deformation of DP600 steel were carried out using a modified Rousselier ductile damage model at different strain rates ranging from 0.1 to 100 s−1. Since the original Rousselier model does not consider any secondary void nucleation or coalescence criteria, it was modified by including a strain-controlled void nucleation function, a coalescence criterion and a void growth acceleration function as the post-coalescence regime identifier. The predicted flow behaviour, the evolution of damage and critical strain and void volume fraction at the onset of coalescence were assessed to evaluate the performance of the proposed model at each strain rate. In addition, X-ray tomography analysis was employed to evaluate the void volume fraction predicted by each void coalescence criterion. The modified Rousselier model showed good agreement with the experimentally determined strain and void volume fraction at the onset of coalescence. Also, it could successfully predict the damage distribution and the final damage geometry of DP600 tensile specimens.

Dual phase (DP) steel sheets are increasingly being used for automotive applications due to relatively high strength and formability. In the current work, Marciniak formability tests were carried out to determine the forming limit curve (FLC) of DP600 steel sheet, and an extended version of Rousselier’s ductile damage model, which accounts for void nucleation, growth and coalescence was used to simulate the tests and predict strain localization and failure for three different strain paths: uniaxial tension, plane strain and biaxial tension. In addition, a combination of flat rolling and uniaxial tension tests were used to generate the extended flow curve of the material. Damage evolution in terms of Rousselier scalar damage variable and void volume fraction was assessed for each simulation condition. The FLC as well as neck and fracture morphologies and geometries were obtained from finite element simulations of the Marciniak tests and compared to experimental results. The sensitivity and dependency of the predicted necking limits, damage distribution and geometry predicted by the Rousselier damage model to the type of hardening model, strain path, void nucleation function and void coalescence criterion are discussed. The modified Rousselier model was shown to successfully predict the FLC, damage distribution and the final damage geometry of DP600 sheets.

In this study, damage mechanisms in AA5182-O aluminium sheets are investigated using quasi-static (QS) Marciniak tests and high strain-rate electro-hydraulic forming (EHF) process. The results confirm that void nucleation, growth and coalescence are the main damage mechanisms of AA5182-O at both high and low strain rates. The EDS analysis suggests that cracking of Al 3 (Fe-Mn) intermetallic particles is the main source of void nucleation, whereas Mg 2 Si particles do not majorly influence void formation. Void growth analysis suggests that specimens deformed under QS conditions contained more voids in areas away from the sub-fracture surface but specimens deformed at high strain rate exhibit more significant rate of void growth close to sub-fracture areas. Void formation is suppressed by increasing the applied energy in EHF. And more significantly, the growth of voids is suppressed due to the high-velocity impact of the sheet against the die which plays an important role in increasing formability of AA5182-O aluminium sheet in EHF process. When the void percentage increase remains less than about 0.5% AA5182-O can be formed safely. However, when the void percentage increases beyond 0.6–0.8% fracture becomes inevitable.

In the current research, the Rousselier ductile damage model was employed to model hardening, plastic instability and damage properties of DP600 during uniaxial tension in a wide range of strain rates (from 0.001 to 1000 s−1). Also, various well-known phenomenological hardening functions, such as Johnson–Cook and KHL as well as a modified version of Johnson-Cook and multiplicative combinations of Voce with other strain-rate hardening functions have been fitted to experimental flow curves via a new combination of non-linear regression and Markov chain Monte Carlo (MCMC) method. The effect of each hardening function on the evolution of the damage parameter, void volume fraction and strain distribution along the gauge length was evaluated throughout the deformation. Also, the onset of instability, geometry of the neck and final fracture were then assessed by comparing the numerical results with experimental data. It is found that the modified JC and Voce-modified JC models can predict the flow behaviour of DP600 more accurately. Additionally, it is shown that the strain hardening rate at large strain levels, as determined by the hardening models, has a considerable effect on the strain map along the specimen, onset of void growth, and progression of damage in the localized area. Keywords: Dual phase steel, Finite element, Modified Johnson–Cook, Markov chain Monte Carlo, Rousselier model, Ductile damage

There is a steady increase in the implementation of dual phase steels in stamped automotive components. Therefore, steel suppliers who develop dual phase steels are interested in predicting the microstructure-properties relationship for optimization of microstructural design. This goal is achievable by micromechanical modeling. The representative volume element (RVE) method has been a popular technique for micromechanical modeling of dual phase steels. It is generally considered that 2D modeling underestimates the flow curves and that 3D modeling predicts the experimental stress-strain curves more accurately. However, much of the research has focused on 2D modeling. This paper develops 3D micromechanical modeling of DP500 and bainite-aided DP600 steels by including statistical quantitative metallography data in the models. More than 3000 grains were analyzed in each steel. Hence, both volume fraction and morphology of martensite were statistically determined. This model predicted the ultimate tensile strength of these two dual phase steels with less than 0.5% error.

In this study, the tensile flow behaviour of AA5182-O sheet was experimentally obtained in different material directions (RD, DD, and TD) at strain rates ranging from 0.001 to 1000 s−1 and predicted by means of both phenomenological models and neural networks (NNs). Constants in Johnson–Cook (JC), Khan–Huang–Liang (KHL), and modified Voce were calculated using genetic algorithm (GA) and linear regression analysis and used to simulate the uniaxial tension tests. Two types of feed-forward back-propagation neural networks were also trained and validated to predict the rheological behaviour of the alloy without the limitations of a mathematical function. The weights and bias values of each network were then used to simulate uniaxial tensile deformation. Subsequently, the results were compared with experimental flow curves and accuracy parameters were calculated. It was found that the modified Voce constitutive equation was able to predict the flow behaviour of AA5182-O with better accuracy than JC and KHL models. Also, the NN was found to be the most accurate method of predicting the anisotropic rate-dependant behaviour of AA5182-O. Keywords: Phenomenological constitutive model, Genetic algorithm, Artificial neural network, 5000 series aluminium alloys, Moderate strain rates

Industrial sheet metal forming processes often involve complex deformation modes and it is necessary to consider nonlinear loading path effects when predicting forming limit curves. Moreover, the yield criterion plays a critical role in the accuracy of predicted forming limits.In this work the MK analysis was modified to relate the initial imperfection factor to a physical property such as the surface roughness, and the orientation of the imperfection was also allowed to vary. This model was used to predict the strain-based and stress-based forming limit curves (FLC and SFLC) of sheet materials that are subject to either linear or non-linear strain paths.Two different yield criteria were employed in this study, Hill's 1948 quadratic yield criterion and Hosford's 1979 non-quadratic yield criterion. The theoretical model was validated by comparing predicted FLC and experimental FLC curves obtained from the literature. FLCs and SFLCs predicted with these two yield criteria were compared for both linear and nonlinear loading paths.Results showed that both the quadratic and non-quadratic yield criteria predict the FLC with acceptable accuracy however on the whole the non-quadratic yield criterion generally provides a slightly better correlation with experimental data, especially on the right side of the FLC. Keywords: Forming limit curve, Formability, Yield criterion, MK analysis, Sheet metal, Loading path

The objectives of this work are to describe the nucleation and growth of voids in DP600 dual phase steel sheets formed along different strain paths, and to investigate the correlation between the micro-mechanisms of damage and the forming limits. DP600 sheet specimens were stretch-formed in uniaxial, plane strain and biaxial tension using the Marciniak–Kuczynski formability test. The evolution of damage in the microstructure was correlated to the formability in each strain path by quantitatively analyzing void density, void area fraction, void aspect ratio and mean void size. Results showed that DP600 steel sheets have less formability in plane strain compared to uniaxial and biaxial tension due to the more rapid elongation and growth of voids. Moreover, the sheets safely deformed to much greater effective strains in biaxial tension due to the smaller mean void size and slower void growth. As proposed by Gurson Gurson (1977) , Tvergaard and Needleman Tvergaard and Needleman (1984) , void volume fraction is an important damage parameter. However, this work shows that void aspect ratio and mean void size also significantly influence the evolution of damage and the formability of dual phase steels. Keywords: Dual phase steel, Microstructure, Strain path, Voids, Forming limit curve

The stress–strain response of some materials, such as advanced high strength steels, during unloading is nonlinear after the material has been loaded into the plastic deformation region. Upon reloading, the response shows a nonlinear elastic response that is different from that in unloading. Therefore, unloading–reloading of these materials forms a hysteresis loop in the elastic region. The Quasi-plastic–elastic model (Sun and Wagoner, 2011) was modified and combined with both isotropic-nonlinear kinematic hardening and two-surface plasticity models to simultaneously describe the nonlinear unloading response and complex cyclic response of sheet metals in the plastic region. The model was implemented as user-defined material subroutines, i.e. UMAT and VUMAT, for ABAQUS/Standard and ABAQUS/Explicit finite element codes, respectively. Uniaxial loading-unloading tests were performed on three common grades of automotive sheet steel: DP600, DP980 and TRIP780 steel. The model was verified by comparing the predicted material response with the corresponding experimental response. Finally, the model was used to predict the springback of a U-shape channel section formed in a plane-strain channel draw process. The results showed that the model was able to considerably improve springback predictions compared to the usual assumption of linear elastic unloading. Keywords: Plasticity, Constitutive model, Nonlinear elastic unloading, Springback

A material model for more thorough analysis of plastic deformation of sheet materials is presented in this paper. This model considers the following aspects of plastic deformation behavior of sheet materials: (1) the anisotropy in yield stresses and in work hardening by using Hill’s 1948 quadratic yield function and non-constant stress ratios which leads to different flow stress hardening in different directions, (2) the anisotropy in plastic strains by using a quadratic plastic potential function and non-associated flow rule, also based on Hill’s 1948 model and r -values, and (3) the cyclic hardening phenomena such as the Bauschinger effect, permanent softening and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening model. Plasticity fundamentals of the model were derived in a general framework and the model calibration procedure was presented for the plasticity formulations. Also, a generic numerical stress integration procedure was developed based on backward-Euler method, so-called multi-stage return mapping algorithm. The model was implemented in the framework of the finite element method to evaluate the simulation results of sheet metal forming processes. Different aspects of the model were verified for two sheet metals, namely DP600 steel and AA6022 aluminum alloy. Results show that the new model is able to accurately predict the sheet material behavior for both anisotropic hardening and cyclic hardening conditions. The drawing of channel sections and the subsequent springback were also simulated with this model for different drawbead configurations. Simulation results show that the current non-associated anisotropic hardening model is able to accurately predict the sidewall curl in the drawn channel sections.

Theoretical and experimental studies have shown that when a compressive stress is applied normal to the sheet surface its formability can be significantly improved. In forming processes such as hydroforming, the normal stress imposed at the surface of the sheet or tube can be very significant in specific locations and it is therefore necessary to account for this stress state when assessing the forming severity of industrial parts. It is also well known that the forming limit curve (FLC) can vary significantly in strain space as a result of non-linear strain paths. In this work, a numerical code based on the Marciniak and Kuczynski (MK) analysis (1967) was developed to predict the FLC of sheet metals by simultaneously accounting for the effects of strain path non-linearity and the normal stress. The FLC predicted with this code can be used to assess the forming severity of parts that are deformed under complex loading conditions in industrial metal forming processes. The predictive FLC model was validated using experimental data in which linear and bi-linear strain paths were applied to different steel and aluminum sheets or tubes under plane stress and three-dimensional stress conditions. This research work has two specific purposes: first, to investigate the influence of the magnitude of the prestrain in bi-linear loading on the sensitivity of the FLC to the normal stress, and second, to study the effect of the normal stress on the path dependence of the FLC. This work showed that, with increasing magnitudes of prestrain the influence of the normal stress on the formability of sheet metal will decrease, regardless of the loading path. Moreover, it was observed that even a significant normal stress will practically not affect the path dependence of the FLC.

AlSi10Mg is an interesting structural material due to its high specific mechanical properties, and good weldability which makes it a good candidate for additive manufacturing by selective laser melting. In this paper, the evolution of pores is visualized and quantified in an AlSi10Mg fabricated by selective laser melting using in-situ tension coupled with X-ray computed tomography. Graphical models were produced in order to visualize the evolution of pores, and the volume fraction, density, and mean diameter of pores were determined at each stage of deformation. Significant coalescence of pores was found during post-uniform elongation. Dimples on the fracture surfaces indicate ductile fracture.

A series of samples from four automotive materials - AKDQ, HSLA50, DP600, and AA6022-T43 - were deformed in a channel draw processes with different levels of draw bead penetration. As a result, varying magnitudes of deformations in plane strain mode and residual stresses were obtained. Through-thickness stress profiles were obtained non-destructively using a novel, high resolution X-ray diffraction technique.

Hydropiercing is an efficient way of piercing holes in mass produced hydroformed parts with complex geometries. By driving piercing punches radially into a hydroformed and fully pressurized tube, holes will be pierced and extruded into the tube-wall. Recent experimental studies have shown that the formability of advanced high strength steel (AHSS) tubes can be increased with the application of internal pressure. In this study, three-dimensional finite element simulations of a tube hydropiercing process of a dual phase steel (DP600) were performed in LS-DYNA, using phenomenological, micromechanical and combined damage criteria. Damage was included in the numerical analysis by applying constant equivalent plastic strain (CEPS), the Gurson-Tvergaard-Needleman (GTN), and the Extended GTN (GTN+JC) model. In order to calibrate the parameters in each model, a specialized hole-piercing fixture was designed and piercing tests were carried out on non-pressurized tube specimens. Of the various ductile fracture criteria, the results predicted with the GTN+JC model, such as the punch load-displacement, the roll-over depth, and the quality of the clearance zone correlated the best with the experimental data.

Under certain conditions, strain rate sensitive materials such as dual phase steels, show formability improvement under high strain rate forming which is known as hyperplasticity. In this research, two commercial dual phase steel sheets, DP500 and DP780, were formed under quasi-static conditions using the Nakazima test and under high strain rate conditions by electrohydraulic forming (EHF) into a conical die. Macro-strains, measured from electro-etched circle grids with an FMTI analyzer, showed remarkable formability improvement in EHF specimens. Micro-strains, i.e., the strains in the ferrite and the martensite, were calculated by quantitative metallography of more than 7000 ferrite grains and 10,500 martensite islands. The goal was to investigate the deformation improvement of the constituents under EHF. Around 20 and 100% deformation improvements were observed in ferrite and martensite, respectively. Furthermore, as a micro-mechanical modeling technique, correlation of the micro-strains with the macro-strains was investigated by applying the mixture rule. Results showed a reasonable correlation between the macro and micro-scale strains; however in banded microstructures, the strain in the martensite should be determined precisely for more accuracy.

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ee ≥ 0.37 for AISI-1012 steel and ee ≥ 0.25 for AA-2008-T4 aluminum).

ASis a relatively new series of Magnesium alloys. The microstructure of this alloy can be improved for semisolid processing. The current research is concerned with the microstructure evolution of AS21 under the strain induced melt activated (SIMA) process. For this purpose, the AS21 alloy is cast and compressed 10-40% at 200 °C. The semisolid heat treatment is completed in a carbonate salt bath at different temperatures between 600-620 °C. The microstructure studies show that there is no favourable microstructure evolution between 600-610 °C. At 615 °C fine globular grains are obtained with the most desired mean grain size and sphericity of 67 µm and 81%, respectively. At 620 °C an undesirable coarsening phenomenon occurs that damages the microstructure globularity. SEM micrographs show that in a successful SIMA processing, the Mg2Si phases are broken into fine particles distributed within the grains and grain boundaries.

The behaviour of dual phase steel tubes at 600, 780, and 980 MPa strength grades undergoing rotary draw bending and pressure sequence hydroforming is examined. Bending was performed using three different bend ratios. Principal strains were measured at several locations on each tube. It was found that the level of strain experienced by the tubes was independent of steel grade. The outcome of the bending process was stochastic in nature; for a given steel grade and bend ratio, some tubes were successfully formed, while others experienced failure. The proportion of failed tubes was found to increase with higher strength steel grades and tighter bend ratios. Metallographic samples from the extrados of the bent tubes revealed that many of the same microstructural features affecting strength and elongation in uniaxial tensile testing, namely martensite banding and non-metallic inclusions, affected dual phase steel tubes undergoing rotary draw bending. Additionally, a nanoindentation technique was investigated to determine the potential for more detailed microstructural characterization.

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948) . The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003) . The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989) . Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.

Pre-bending is a critical process required prior to hydroforming. The bending has an effect on the tube thickness and strain which will use up a portion of the formability of the as-received tube. To compensate for this loss of formability, a multi-objective optimization method was applied to improve the hydroforming process after pre-bending. A multi-objective genetic algorithm (MOGA) and Kriging surrogate model were used to optimize the loading path. The Kriging model was used to replace the finite element simulation in constraint handling. The optimal loading parameters in the hydroforming process were obtained for a tube that was previously bent 90°, and showed an improvement in reducing the corner radii of the part at the extrados and intrados of the bend (8.73 mm and 11.24 mm for the extrados and intrados of the bend, respectively). The corresponding corner fill expansion (CFE) was improved by 16.7% (or 1.79 mm) compared to the maximum expansion of 10.73 mm obtained experimentally.

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ3), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (\( \beta = {{{{\sigma_3}}} \left/ {{{\sigma_1}}} \right.} \)). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.

The forming limit curve (FLC), a plot of the limiting principal surface strains that can be sustained by sheet metals prior to the onset of localized necking, is useful for characterizing the formability of sheet metal and assessing the forming severity of a drawing or stamping process. Both experimental and theoretical work reported in the literature has shown that the FLC is significantly strain-path dependent. In this paper, a modified Marciniak and Kuczynski (MK) approach was used to compute the FLC in conjunction with two different work-hardening models: an isotropic hardening model and a mixed isotropic–nonlinear kinematic hardening model, which is capable of describing the Bauschinger effect. Predictions of the FLC using the MK analysis have been shown to be dependent on the shape of the initial yield locus and on its evolution during work hardening; therefore the hardening model has an influence on the predicted FLC. In this investigation, published experimental FLCs of AISI-1012 low carbon steel and 2008-T4 aluminum alloy sheets that were subjected to various nonlinear loading paths were compared to predictions using both hardening models. The predicted FLCs were found to correlate quite well with experimental data and the effects of strain path changes and of the hardening model on predicted FLCs are discussed.

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain (\( \bar{\varepsilon } \geqslant 0.35 \) for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.

A semi-implicit integration scheme was used to implement the Yoshida–Uemori two-surface model into the finite element method. Hill’s quadratic yield function was employed to account for the orthotropic behaviour of the metal sheet. The model was used to predict the cyclic simple shear response of DP-600. In order to evaluate the capability of the model for sheet metal forming, the model was used to simulate both the forming stage of a channel draw process in the presence of drawbeads and the subsequent springback stage. The results show that the model predicts the springback profile very well, especially at deeper drawbead penetrations.

The return mapping procedure was used to develop an algorithm for numerical implementation of Yoshida–Uemori two-surface plasticity model into a finite element program. A fully implicit integration scheme is utilized to integrate all plasticity equations. The algorithm was employed to develop user material subroutine (UMAT and VUMAT) for both ABAQUS-Standard and ABAQUS-Explicit codes. The numerical algorithm is quite general and is not limited to any particular yield function. However, as an example, the Yld2000-2d yield function was implemented in the subroutines in order to take the anisotropy of metal sheets into account. Finally, the subroutines were used to simulate the springback of a U-shape channel section. The channels were formed using two different drawbead penetrations and two different sheet materials, i.e. HSLA and AA6022-T43. The forming process and subsequent springback stage was simulated using the same yield function and three different hardening laws: (a) isotropic hardening, (b) a combined isotropic-nonlinear kinematic hardening and (c) the Yoshida–Uemori two-surface model. A comparison of the experimental and predicted channel sidewall profiles shows that the Yoshida–Uemori model generally improves the springback prediction compared to isotropic hardening and combined isotropic-nonlinear kinematic hardening.

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.

The objective of this work is to predict the springback of Numisheet’05 Benchmark#3 with different material models using the commercial finite element code ABAQUS. This Benchmark consisted of drawing straight channel sections using different sheet materials and four different drawbead penetrations. Numerical simulations were performed using Hill's 1948 anisotropic yield function and two types of hardening models: isotropic hardening (IH) and combined isotropic-nonlinear kinematic hardening (NKH). A user-defined material subroutine was developed based on Hill's quadratic yield function and mixed isotropic-nonlinear kinematic hardening models for both ABAQUS-Explicit (VUMAT) and ABAQUS-Standard (UMAT). The work hardening behavior of the AA6022-T43 aluminum alloy was described with the Voce model and that of the DP600, HSLA and AKDQ steels with Hollomon's power law. Kinematic hardening was modeled using the Armstrong–Frederick nonlinear kinematic hardening model with the purpose of accounting for cyclic deformation phenomena such as the Bauschinger effect and yield stress saturation which are important for springback prediction. The effect of drawbead penetration or restraining force on the springback has also been studied. Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior. Comparisons between simulation results and experimental data showed that the IH model generally overestimated the predicted amount of springback due to higher stresses derived by this model. On the other hand, the NKH model was able to predict the springback significantly more accurately than the IH model.

A biaxial testing apparatus was used to investigate the elastic-plastic behaviour of an 1145 aluminum sheet alloy. Flat cruciform specimens were deformed up to effective strains of approximately 0.15 in biaxial stretching, along seven different proportional strain paths. A finite element analysis of each test was carried out using four different phenomenological models of anisotropic plasticity. An iterative procedure was coupled with the numerical analyses in order to determine the anisotropic parameters in the various yield functions and thus obtain best fits to these functions. This method of numerically analyzing cruciform specimens leads to fairly accurate biaxial flow curves, as well as plastic work contours. The results are also compared with crystal plasticity simulations using a Taylor-type model.